Whats wrong with my "snowglobe" object and/or lighting?
The viewport is in 'lookdev' mode. That means there is a hdri image as environment lighting. You don't have it while rendering. Lookdev is a builtin environment lighting for test purpose. Should set up an equivalent environment texture to render it. You can find some available hdri images in your installation directory in 'C:\Program Files\Blender Foundation\Blender\2.80\datafiles\studiolights\world'.

To setup that, you basically need to use this setting where 'forest.exr' is found in the directory indicated above:



But if you just render it, you'll see it in the rendering:



So you need to enhance the world node tree a bit (I've found no option to hide it automatically in Eevee).



This setting says: when a light ray is directly to the camera, then use the second 'background' node which is set with a transparent color.

So you can render this:



Here is the blend file (you may need to reopen/reload the hdri image to make it work in your environment):